Abstract
We propose a new path integral based interpretation of quantum field theory (QFT). In our interpretation, QFT is the continuous approximation of a more fundamental, discrete graph theory (theory X) whereby the transition amplitude Z is not viewed as a sum over all paths in configuration space, but measures the symmetry of the differential operator and source vector of the discrete graphical action. We propose that the differential operator and source vector of theory X are related via a self-consistency criterion (SCC) based on the identity that underwrites divergence-free sources in classical field theory, i.e., the boundary of a boundary principle. In this approach, the SCC ensures the source vector is divergence-free and resides in the row space of the differential operator.
Accordingly, the differential operator will necessarily have a non-trivial eigenvector with eigenvalue zero, so the SCC is the origin of gauge invariance. Factors of infinity
associated with gauge groups of infinite volume are excluded in our approach, since Z is restricted to the row space of the differential operator and source vector. We show it is possible that the underlying theory X, despite being discrete, is the basis for exact
Poincaré invariance. Using this formalism, we obtain the two-source transition amplitude over a (1+1)-dimensional graph with N vertices fundamental to the scalar Gaussian theory and interpret it in the context of the twin-slit experiment to provide a unified account of the Aharonov-Bohm effect and quantum non-separability (superposition and entanglement) that illustrates our ontic structural realist alternative to problematic particle and field ontologies. Our account also explains the need for regularization and
renormalization, explains gauge invariance and largely discharges the problems of inequivalent representations and Haag’s theorem. This view suggests corrections to general relativity via modifications to its graphical counterpart, Regge calculus. We conclude by presenting the results of our modified Regge calculus approach to Einstein-de Sitter cosmology where we produced a fit to the Union2 Compilation data for type Ia supernovae rivaling that of the concordance model (ΛCDM), but without having to invoke dark energy or accelerated expansion.
PACS: 03.65.Ca; 03.65.Ta; 03.65.Ud; 11.15.-q
Key Words: graph theory, path integral, gauge invariance, quantum field theory, transition amplitude
1
Department of Physics, Elizabethtown College, Elizabethtown, PA 17022, [email protected] 2Department of Philosophy, Elizabethtown College, Elizabethtown, PA 17022, [email protected] 3 Department of Philosophy, University of Maryland, College Park, MD 20742
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1. INTRODUCTION
1.1 Foundational Problems of Quantum Field Theory. When it comes to quantum field theory (QFT) some have stressed that the conceptual problems besetting non-relativistic quantum mechanics (NRQM) remain the central concerns(1), while others stress that QFT exacerbates some of the interpretive problems of NRQM and possesses foundational problems all its own(2). Some (especially physicists) have stressed that QFT is the greatest and most explanatory intellectual achievement of modern science(3), while others believe QFT is “much more a set of formal strategies and mathematical tools than a closed theory(4).” Of course on both counts, both sides are right. In addition to the problems of NRQM, an interpretation must address concerns unique to QFT, e.g., notorious problems with particle and field ontologies and renormalization, how to interpret gauge invariance and the Aharonov-Bohm effect (AB effect), the problem of inequivalent representations, and explaining the effectiveness of the interaction picture and perturbation theory in light of Haag’s theorem. As for progress in this area, Healey notes(5), “no consensus has yet emerged, even on how to interpret the theory of a free, quantized, real scalar field.” And(6), “There is no agreement as to what object or objects a quantum field theory purports to describe, let alone what their basic properties would be.”
Those who emphasize the incompleteness of QFT over its successes often focus on the many ad hoc and, for some, troubling “fixes” involved in the practice of QFT1. For example, since QFT is independent of overall factors in the transition amplitude, such factors are simply “thrown away” even when these factors are infinity as is the case when the volume of the gauge symmetry group in Fadeev-Popov gauge fixing is infinite(7). And, in the process of renormalization one must “tweak” parameters in the Lagrangian so they remain finite under regularization(8). QFT has triumphed empirically, but virtually all agree that it is not a fundamental theory because it does have a limited domain of applicability, viz., it does not deal with particle interactions at ranges where gravity becomes important. It might be that the Standard Model plus the gravitational field is fundamental(9), but most physicists assume there exists an underlying, unified theory
1 We are focusing on the “textbook variant of QFT.” Fraser, D.: Quantum Field Theory:
Underdetermination, Inconsistency, and Idealization. Philosophy of Science 74, 536-565 (October 2009). In particular, we are concerned primarily with QFT as applied to particle physics.
called quantum gravity which would naturally justify the ad hoc fixes employed in QFT and tell us how to handle the particle interactions where gravity is deemed relevant(10).
Clearly, QFT is in need of more philosophical attention. As Glashow stated(11), “in a sense it really is a time for people like you, philosophers, to contemplate not where we’re going, because we don’t really know and you hear all kinds of strange views, but where we are. And maybe the time has come for you to tell us where we are.” Rovelli goes further stating(12), “As a physicist involved in this effort, I wish that the philosophers who are interested in the scientific description of the world would not confine themselves to commenting and polishing the present fragmentary physical theories, but would take the risk of trying to look ahead.” Consequently, we propose a new ontology and commensurate path integral account of “theory X” underlying QFT2.
1.2 Ontic Structural Realism in a Blockworld: The Graphical, Quantum and Classical. Our account of spacetime and matter is very much in keeping with Rovelli’s intuition that(13):
General relativity (GR) altered the classical understanding of the concepts of space and time in a way which...is far from being fully understood yet. QM challenged the classical account of matter and causality, to a degree which is still the subject of controversies. After the discovery of GR we are no longer sure of what is spacetime and after the discovery of QM we are no longer sure of what matter is. The very distinction between space-time and matter is likely to be ill-founded....I think it is fair to say that today we do not have a consistent picture of the physical world. [italics added]
Our ontological account of quantum physics is conceptually challenging but, succinctly, it is a form of ontic structural realism in a blockworld setting (4D)3 with a co-determining amalgam of space, time and matter that we call “spacetimematter.” As with GR,
topological and geometric properties are fundamental, but on our view matter cannot be separated at all from spacetime (unlike GR with its vacuum solutions), so matter also gets a geometric treatment. We will briefly unpack this description in the remainder of this
2 Here we follow the possibility articulated by Wallace (p 45) that, “QFTs as a whole are to be regarded only as approximate descriptions of some as-yet-unknown deeper theory,” which he calls “theory X.” Wallace, D.: In defence of naiveté: The conceptual status of Lagrangian quantum field theory. Synthese 151, 33-80 (2006).
3
For the reader with an aversion to 4Dism (blockworld), we are simply saying topological and geometric facts that encompass the entire history of physical systems are deeper than dynamical or mechanical facts.
subsection, but we don’t expect the reader will fully appreciate or totally understand this ontology until reading the commensurate formalism in sections 2 & 3, because our account is metaphysically and methodologically perverse by the lights of what we will call dynamism or the dynamical bias. Fundamental theories of physics (M-theory, loop quantum gravity, causets, etc.) may deviate from the norm by employing radical new fundamental entities (branes, loops, ordered sets, etc.), but the game is always dynamical, broadly construed (vibrating branes, geometrodynamics, sequential growth process, etc.). As Healey puts it(14):
Physics proceeds by first analyzing the phenomena with which it deals into various kinds of systems, and them ascribing states to such systems. To classify an object as a certain kind of physical system is to ascribe certain, relatively stable, qualitative intrinsic properties; and to further specify the state of a physical system is to ascribe to it additional, more transitory [time dependent], qualitative intrinsic properties….A physical property of an object will then be both
qualitative and intrinsic just in case its possession by that object is wholly
determined by the underlying physical states and physical relations of all the basic systems that compose that object.
If one takes it on faith that dynamical explanation is fundamental (however far from ordinary experience and classical physics it might be), it may be impossible to take us seriously, maybe even impossible to clearly envision what we are suggesting. Our
ontology and our fundamental methodology violate every tenet of dynamism. Indeed, we will argue that the incompatibility of quantum physics and general relativity is really pointing to the relative failure of dynamism at more fundamental “levels.”
Our violation of dynamism is in accord with ontic structural realism(15) (OSR): Ontic structural realists argue that what we have learned from contemporary physics is that the nature of space, time and matter are not compatible with standard metaphysical views about the ontological relationship between
individuals, intrinsic properties and relations. On the broadest construal OSR is any form of structural realism based on an ontological or metaphysical thesis that inflates the ontological priority of structure and relations.
More specifically, our version of OSR (called Relational Blockworld—RBW (16)) claims that(17) “The relata of a given relation always turn out to be relational structures
themselves on further analysis.” Note that OSR does not claim there are relations without relata, just that the relata are not individuals (e.g., things with primitive thisness and
intrinsic properties), but always ultimately analyzable as relations as well. As will be apparent in section 3, there is no infinite regress of relata and relations in our graphical approach, because a boundary operator on the vector of links (fundamental relations) produces a very intuitive, but not tautological, characterization for the vector of nodes (relata for the links). OSR already violates the dynamical bias by rejecting things with intrinsic properties and their dynamics as fundamental building blocks of reality—the world isn’t fundamentally compositional—the deepest conception of reality is not one in which we decompose things into other things at ever smaller length and time scales. Unfortunately for dynamism, we must further exacerbate this violation by applying OSR to a blockworld.
The blockworld perspective (the reality of all events past, present and future including the outcomes of quantum experiments) is suggested by the relativity of simultaneity in special relativity or, more generally, the lack of a preferred spatial foliation of spacetime in GR, and even by quantum entanglement according to some of us(18). Geroch writes(19):
There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes. In particular, one does not think of particles as moving through space-time, or as following along their world-lines. Rather, particles are just in space-time, once and for all, and the world-line represents, all at once, the complete life history of the particle.
When Geroch says that “there is no dynamics within space-time itself,” he is not denying that the mosaic of the blockworld possesses patterns that can be described with dynamical laws. Nor is he denying the predictive and explanatory value of such laws. Rather, given the reality of all events in a blockworld, dynamics are not “event factories” that bring heretofore non-existent events (such as measurement outcomes) into being; fundamental dynamical laws that are allegedly responsible for
discharging fundamental “why” questions in physics are not brute unexplained explainers that “produce” events on our view. Geroch is advocating for what philosophers call Humeanism about laws. Namely, the claim is that relatively fundamental dynamical laws are descriptions of regularities and not the brute
explanation for such regularities. His point is that in a blockworld, Humeanism about laws is an obvious position to take because everything is just “there” from a “God’s
eye” (Archimedean) point of view. There is a caveat, however. In the relational reality of RBW, there can be no “God’s eye” point of view because “observers” have to be part of that which they observe—themselves relations in a relational network. Consequently, in section 2, we argue for an OSR blockworld characterized as spacetimematter, as opposed to the spacetime + matter picture of current physics.
To formalize spacetimematter and provide a basis for quantum physics we will use graphical relations to self-consistently4 co-construct space, time and sources5 (matter) in a graphical fashion (theory X). There are two immediate conceptual consequences to spacetimematter. First, there is no “empty spacetime” so GR, which contains vacuum solutions, cannot be a fundamental theory of physics per theory X. We will speculate briefly on how GR must be “corrected” in section 5. In essence we claim that GR phenomena are only approximately separable in a statistical sense to be specified, and therefore GR is applicable only when its approximation of
“separability” holds. As Healey notes(20), “By contrast, classical general relativity is separable, since all the qualitative intrinsic physical properties it ascribes on a loop do supervene on qualitative intrinsic physical properties assigned on (infinitesimal neighborhoods of) space-time points on that loop.” On the spacetime + matter picture it is common to try and square quantum non-separability with the separability of GR. This has proven to be problematic thus far. We resolve this problem with our
spacetimematter theory wherein the non-separability of quantum states in Hilbert space and Healey’s characterization of non-separability in terms of the relations between spacetime points (such as EPR correlations) get a unified explanation.
Second, there are no “quantum entities” with “quantum states” (of any sort) emitted by the Source, moving through the various pieces of experimental equipment (e.g., beam splitters, mirrors) and impinging on the detector(s) to cause experimental outcomes in quantum experiments. Space, time and sources are co-constructed (a fusion or unity) to represent the relevant relationships comprising the various pieces of experimental equipment (OSR) from an experiment’s initiation to its termination
4 Our form of self-consistency is topological, i.e., it is characterized via boundary operators in the spacetime chain complex of our spacetimematter graph.
5 We use the word “source” as in QFT, i.e., to mean “particle sources” or “particle sinks” (creation or annihilation events, respectively). When we want to specify “a source of particles” we will use “Source.”
(blockworld); past, present and future are co-constructed as well, there are no dynamical entities or dynamical laws in our fundamental formalism. As we shall soon explain, spacetimematter underwrites quantum non-separability (superposition and entanglement) in a kinematical fashion. Accordingly, all dynamical explanation supervenes on, and is secondary to, non-dynamical topological facts about the graph world.
Consequently, fundamental explanation is in terms of a global, adynamical organizing principle. Thus, ultimate explanation in physics is not in terms of some
thing or dynamical entity (obeying a new dynamical equation) “at the bottom” conceived at higher energies and smaller spatiotemporal scales, begging for justification from something at some yet “deeper” scale, but self-consistency writ large for the explanatory “process” as a whole. As we shall see, this goes well beyond consistency as typically conceived by physicists. Self-consistency writ large is extremum thinking writ large, which truly transcends and underwrites the
dynamical perspective. Mathematically speaking, the topological characterization of self-consistent spacetimematter at the graphical level is mirrored by the resulting geometric classical field theory at the classical level.
In short, distributions of spatiotemporal geometric relations over the (topological) graphical realm (Figure 1 Figure 2) are averaged to obtain the spacetime geometry for the unity of spacetimematter of the classical realm (Figure 3). Classical equations of motion are given in terms of this “average spacetime geometry” for the unity of spacetimematter such that the standard
spacetime + matter picture obtains as a statistical approximation. A graph (Figure 1) overlaid with a particular spatiotemporal geometric distribution (Figure 2) can result in geometrically localizable subsets, which we call “Clusters” in Figure 2. There are several different geometric versions of a Cluster that are consistent with a particular classical Object (Figure 3), analogous to the many different velocity distributions for the molecules of a gas that give rise to the same temperature and pressure per
statistical mechanics. If one wants to explore specific spatiotemporal geometric relations (specific line segments in Figure 2) in a particular distribution over the graph (specific trial in the experiment), one is doing quantum physics (Figure 4).
This ontology of spatiotemporal geometry over spacetimematter graphs can be described in terms of the distributions of individual geometric relations (quantum physics) or it can be described approximately in terms of the averages of the distributions of individual geometric relations (classical physics). Obviously, the (average) spacetime + matter approximation becomes more accurate as the number of geometric relations increases.
Probably the most important aspect of the RBW ontology for the
interpretation of quantum physics is that there are no “quantum Clusters,” so there are no “quantum Objects,” i.e., all Objects are classical and quantum physics is an exploration of their relational “composition” (Figure 4). This is in stark contrast to those interpretations of quantum physics which employ dynamical ontological
constituents of the essentially quantum realm (particles, waves, wave-functions, fields, etc.) with their strange non-commutative properties and struggle to somehow compose or realize the essentially classical realm of dynamical ontological
constituents with commutative properties. Thus, there simply is no possibility of a measurement problem(21) on our view (a problem driven by taking quantum dynamics realistically), and quantum non-separability is ultimately explained kinematically by the unity of spacetimematter. In section 3, we will show how the fusion of spacetimematter in this approach explains the interference pattern of the twin-slit experiment without invoking “quantum entities” moving through space as a function of time to “cause” detector events. But, before jumping into the formalism, we want to provide a conceptual primer.
Methodologically, we start with a graph and use boundary operators in its spacetime chain complex to provide a topological representation of the relations under investigation in a particular experiment. We use this topological characterization to create a partition function for the ensemble of possible geometric relations over the spacetimematter graph. Essentially, this partition function provides a measure of the graph’s ability to accommodate various spacetime geometries for its unity of
spacetimematter6. So, the equipment in a particular quantum experiment, understood in
6 Technically speaking, we use a discrete path integral over graphs with a Wick-rotated action in the transition amplitude. All this will be explained in sections 2 & 3.
the context of an “average spacetime geometry” (Figure 3), is idealized as the instantiation by the graphical spacetimematter (Figure 1) of some particular
spatiotemporal geometric distribution in the ensemble (Figure 2), where one can have a different distribution for each trial of the experiment (again, there can be many different spatiotemporal geometric distributions consistent with a particular average spacetime + matter experimental configuration). The experimental outcome then reflects a specific
spatiotemporal geometric relation in the distribution of that particular trial (Figure 4). Thus, the partition function is used to compute the probability of finding a specific spatiotemporal geometric relation (representing a particular experimental outcome) in the conduct of the experiment. The most probable of these specific outcomes is given by the extremum of the probability function and, since the most probable value is the average value in our Gaussian distribution, we recoverclassical equations of motion in terms of the “average spacetime geometry” for the unity of spacetimematter. As will be seen, the manner by which the boundary operators in the spacetime chain complex of the graph give rise to its partition function is mirrored precisely in the classical equations of motion. As we explain in section 5, the classical result is a sort of modified Regge calculus7, which obviously suggests a bridge from theory X for spacetimematter to its continuous, separable, statistical approximation of GR for spacetime + matter.
Given Figures 1-4 and the explanation immediately above, it should be clear how the ontology of spacetimematter gives rise to quantum non-separability. The unity of spacetimematter gives the separable spacetime + matter on average as an approximation to situations involving large numbers of geometric relations. But, it is possible to
construct (quantum) experiments that reveal individual relations between classical Objects which then appear as non-separable outcomes in the context of the “average spacetime geometry” over spacetime + matter. Thus, quantum non-separability will be “mysterious” if one believes (erroneously) that the separable spacetime + matter is fundamental, rather than recognizing it as a mere statistical approximation to what is truly fundamental, i.e., the unity of spacetimematter. [More on this in section 5.]
7 Regge calculus is a discrete approximation to general relativity where the discrete counterpart to Einstein’s equations is obtained from the least action principal on a 4D graph. This generates a rule for constructing a discrete approximation to the spacetime manifold of GR using 4D graphical “tetrahedra” called “simplices” (Figure 10). For more information, see Chap 42 of Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San Francisco (1973).
Thus, the payoff for an OSR blockworld ontology (with its commensurate methodology) that violates the dynamical bias is a unified picture of physics (theory X) that resolves the conceptual, foundational and technical issues of quantum
physics. This is in accord with Smolin’s prediction(22) that, “The problem of quantum mechanics is unlikely to be solved in isolation; instead, the solution will probably emerge as we make progress on the greater effort to unify physics.” Unfortunately, the mathematical counterpart to this extremely counterintuitive ontology is equally obscure, i.e., a discrete path integral over graphs.
1.3 Overview of Paper. We understand the reader may not be familiar with the path integral formalism, as Healey puts it(23), “While many contemporary physics texts present the path-integral quantization of gauge field theories, and the mathematics of this
technique have been intensively studied, I know of no sustained critical discussions of its conceptual foundations.” Therefore, we begin in section 2 with an overview and
interpretation of the path integral formalism. Immediately after we introduce and interpret the path integral formalism, we motivate our use of a discrete path integral approach to theory X to include the self-consistency criterion (SCC) responsible for the
co-construction of space, time and matter. The SCC is based on the boundary of a boundary equals zero (∂∂ = 0), responsible for the divergence-free nature of the stress-energy tensor in classical physics8.The SCC provides the rule by which boundary operators in the spacetime chain complex of the graph “provide a topological representation of the relations under investigation in a particular experiment.”
In section 3, we provide the mathematical details of theory X via our discrete path integral formalism over graphs, explaining how it yields quantum physics and classical physics in its continuum wake. Using this formalism, we obtain the two-source transition amplitude over a (1+1)-dimensional graph with N vertices fundamental to the scalar Gaussian theory, and interpret it in the context of the twin-slit experiment. Having formally composed our OSR blockworld, we address various conceptual and technical issues associated with QFT in section 4. Specifically, we provide an OSR alternative to problematic particle and field ontologies that also explains the need for regularization and
8 A divergence-free stress-energy tensor characterizes the conservation of momentum and energy in classical physics.
renormalization, explain gauge invariance, provide a unified account of the Aharonov-Bohm effect and quantum non-separability, and largely discharge the problem of
inequivalent representations. We will also speculate on how our graphical theory X might provide a basis for exact Poincaré invariance, which includes Lorentz invariance –
typically a problem for discrete lattice theories9. We conclude section 4 with a brief explanation of why Haag’s theorem creates problems for the interaction picture according to theory X. In section 5, we provide a summary using the Maxwell and Einstein-Hilbert actions as examples, and present the results of our modified Regge calculus approach to Einstein-de Sitter cosmology which produced a sum of squares error (SSE) in fitting the Union2 Compilation data for type Ia supernovae of 1.77. This result rivals the best fit (SSE = 1.79) of this same data by the concordance model (ΛCDM), but without having to invoke dark energy or accelerated expansion (Figure 11).
2. THE DISCRETE PATH INTEGRAL FORMALISM AND RBW
In this section we provide an overview and interpretation of the path integral approach, showing explicitly how we intend to use “its conceptual foundations.” We employ the discrete path integral formalism because it embodies a 4Dism of the sort outlined above that allows us to model spacetimematter. For example, the path integral approach is based on the fact that(24) “the [S]ource will emit and the detector receive10,” i.e., the formalism deals with Sources and sinks as a unity while requiring a description of the experimental process from initiation to termination (blockworld). By assuming the discrete path integral is fundamental to the (conventional) continuum path integral, we have a graphical basis for the co-construction of time, space and quantum sources via a self-consistency criterion (SCC). We will show in section 3 how the graphical amalgam of spacetimematter is the basis for quantum and classical physics.
2.1 Path Integral in Quantum Physics.In the conventional path integral formalism(25) for NRQM one starts with the amplitude for the propagation from the initial point in
configuration space qI to the final point in configuration space qF in time T via the unitary
9 In lattice gauge theory, spacetime is modeled as a hypercubic lattice in 4-dimensional Euclidean space. One obtains rotationally invariant QFT in the limit as the lattice spacing goes to zero, and this gives Lorentz invariance after Wick rotation. However, one does not have the full rotational invariance on the discrete lattice, so lattice theories which are to remain discrete typically have problems with exact Lorentz invariance.
10 The path integral formalism requires both an emission event and a reception event; the formalism was motivated by the idea of treating advanced and retarded potentials equally.
operator eiHT, i.e.,
I iHT
F e q
q . Breaking the time T into N pieces δt and inserting the
identity between each pair of operators eiHt via the complete set
dqq q 1 we have I t iH t iH N t iH N N t iH F N j j I iHT F e q dq q e q q e q q e q q e q q
1 1 1 2 2 1 1 1 With (ˆ) 2 ˆ2 q V m pH and δt 0 one can then show that the amplitude is given by
T I iHT F e q Dq t i dtL q q q 0 ) , ( exp ) ( (1) where ( ) 2 1 ) , (q q mq2 V qL . If q is the spatial coordinate on a detector transverse to the line joining Source and detector, then
1 1 N j
can be thought of as N-1 “intermediate” detector surfaces interposed between the Source and the final (real) detector, and
dqjcan be thought of all possible detection sites on the jth intermediate detector surface. In the continuum limit, these become
Dq(t)which is therefore viewed as a “sum over all possible paths” from the Source to a particular point on the (real) detector, thus the term “path integral formalism” for conventional NRQM is typically understood as a sum over “all paths through space.”To obtain the path integral approach to QFT one associates q with the oscillator displacement at a particular point in space (V(q) = kq2/2). In QFT, one takes the limit δx 0 so that space is filled with oscillators and the resulting spatial continuity is accounted for mathematically via qi(t) q(t,x), which is denoted φ(t,x) and called a “field.” TheQFT amplitude (denoted “Z”) then looks like
exp 4 (,) xL d i D Z (2) where
( ) 2 1 ) , ( d 2 V L . Impulses J are located in the field to account for particle creation and annihilation; these J are called “sources” in QFT and we have
( ) ( , ) ( , ) 2 1 ) , ( d 2 V J t x t x) , ( ) , ( 2 1 ) , ( D J t x t x
L , where D is a differential operator. In its discrete form (typically, but not necessarily, a hypercubic spacetime lattice), D K(a difference matrix), J(t,x) J (each component of which is associated with a point on the
spacetime lattice11) and φ Q (each component of which is associated with a point on the spacetime lattice). The discrete counterpart to Eq. (2) is then(26)
dQ dQ i Q K Q iJ Q Z N 2 exp ... ... 1 (3). In conventional quantum physics, NRQM is understood as (0+1)-dimensional QFT.2.2 Our Interpretation of the Path Integral in Quantum Physics. We agree that NRQM is to be understood as (0+1)-dimensional QFT, but point out this is at conceptual odds with our derivation of Eq. (1) when
Dq(t)represented a sum over all paths in space, i.e., when q was understood as a location in space (specifically, a location along a detector surface). If NRQM is (0+1)-dimensional QFT, then q is a field displacement at a single location in space. In that case,
Dq(t)must represent a sum over all field values at a particular point on the detector, not a sum over all paths through space from the Source to a particular point on the detector. So, how do we relate a point on the detector (sink) to the Source?In answering this question, we now explain a formal difference between
conventional path integral NRQM and our proposed approach: roughly we are connecting discrete sourcesJ, where one part of J is used for the Source and the other part of Jis used for the detector click (sink). Instead of δx 0, as in QFT, we assume δx is
measureable for (such) NRQM phenomenon. More specifically, we propose starting with Eq. (3)12 whence (roughly) NRQM obtains in the limit δt 0, as in deriving Eq. (1), and QFT obtains in the additional limit δx 0, as in deriving Eq. (2). The QFT limit is well
11 Part ofJrepresents particle Sources the other part represents particle sinks in the conventional view of path integral QFT so that field disturbances emanate from one source location (Source) and are absorbed at another source location (sink). In particle physics, these field disturbances are the particles. We will keep the partition of Jinto Sources and sinks in our theory X, but there will be no disturbance (or any “thing” else) propagating between them because, as we shall show, there will be no medium (field) to be
“disturbed” between the discrete set of sources.
understood as it is the basis for lattice gauge theory and regularization techniques, so one might argue that we are simply clarifying the NRQM limit where the path integral formalism is not widely employed. However, again, we are proposing a discrete starting point13 for theory X, as in Eq. (3).
2.3 Discrete Path Integral is Fundamental. The version of theory X we propose is a discrete path integral over graphs, so Eq. (3) is not a discrete approximation of
Eqs. (1) & (2), but rather Eqs. (1) & (2) are continuous approximations of Eq. (3). In the arena of quantum gravity it is not unusual to find discrete theories(27) that are in some way underneath spacetime theory and theories of “matter” involving dynamical entities such as QFT, e.g., causal dynamical triangulations(28), quantum graphity(29) and causets(30). While these approaches are interesting and promising, the approach taken here for theory X will look more like Regge calculus quantum gravity (see Bahr & Dittrich (31) and references therein for recent work along these lines).
Placing a discrete path integral at bottom introduces conceptual and analytical deviations from the conventional, continuum path integral approach. Conceptually, Eq. (1) of NRQM represents a sum over all field values at a particular point on the detector, while Eq. (3) of theory X is a mathematical machine that measures the “symmetry” (strength of stationary points) contained in the core of the discrete action
J K
2 1
(4). This core or actional yields the discrete action after operating on a particular vector Q
(field). The actional represents a fundamental, 4D description of the experimental
arrangement and Z is a measure of its symmetry14. For this reason, and because transition
amplitude connotes a dynamical process, we prefer to call Z the symmetry amplitude of the 4D experimental configuration. Since Qis only an integration variable, fields have no ontic significance at this fundamental level – they are merely part of the computational device for measuring the symmetry of the actional (representing what is ontically
significant at the fundamental level). Analytically, because we are starting with a discrete
13 That discrete spacetime is fundamental while “the usual continuum theory is very likely only an approximation” is, of course, an old idea. See, for example, arguments in Feinberg, G., Friedberg, R., Lee, T.D., and Ren, H.C.: Lattice Gravity Near the Continuum Limit. Nuclear Physics B245, 343-368 (1984). 14 In its Euclidean form, which is the form we will use, Z is a partition function.
formalism, we are in position to mathematically explicate trans-temporal identity, whereas this process is unarticulated elsewhere in physics (as elaborated immediately below). As we will now see, this leads to our proposed self-consistency criterion (SCC) underlying Z.
2.4 Time, Space & Discrete Quantum SourcesJ. The NRQM limit δt 0 of Eq. (3) results in a spatially discrete distribution of “interacting” sources Ji(t) and illustrates a key aspect of the RBW ontology, i.e., what is typically understood as “interaction” in
quantum physics is modeled without mediating waves, particles, etc., traveling through intervening space (in fact, there is no medium either, i.e., field, between sources Ji(t)). The spatiotemporally discrete formalism also illustrates nicely how NRQM tacitly
assumes an a priori process of trans-temporal identification, J Ji(t) as δt 0. Indeed, there is no principle which dictates the construct of diachronic entities fundamental to the formalism of dynamics in general – these objects are “put in by hand” throughout
physics. When Albrecht and Iglesias(32) allowed time to be an “internal variable” after quantization, as in the Wheeler-DeWitt equation, they found “there is no one set of laws, but a whole library of different cosmic law books(33).” They called this the “clock
ambiguity.” In order to circumvent this “arbitrariness in the predictions of the theory” they proposed that “the principle behind the regularities that govern the interaction of entities is … the idea that individual entities exist at all(34).” Albrecht and Iglesias characterize this as “the central role of quasiseparability.”
Similarly, the RBW approach requires a fundamental principle (∂∂ = 0) whence the trans-temporal identity employed tacitly in NRQM and all dynamical theories. Our discrete (graphical) starting point provides a topological basis for sourcesJ, space and time. Clearly, the processJJi(t) is an organization of the set Jon two levels – there is the split of the set into i subsets, one for each source, and there is the ordering t over each subset. The split represents space, the ordering represents time and the result is (trans-temporal) objecthood. In this sense, space, time and sourcesJare relationally
co-constructed in our formalism. Consequently, we believe the articulation of the otherwise tacit construct of dynamical entities has a mathematical counterpart fundamental to the
action, viz., the boundary of a boundary principle, ∂∂ = 0, at the fundamental level15.This is in accord with Toffoli’s belief that there exists a mathematical tautology fundamental to the action(35):
Rather, the motivation is that principles of great generality must be by their very nature trivial, that is, expressions of broad tautological identities. If the principle of least action, which is so general, still looks somewhat mysterious, that means we still do not understand what it is really an expression of—what it is trying to tell us.
2.5 Self-Consistency Criterion. Our use of a self-consistency criterion is not without precedent, as we already have an ideal example in Einstein’s equations of GR. Momentum, force and energy all depend on spatiotemporal measurements (tacit or explicit), so the stress-energy tensor cannot be constructed without tacit or explicit
knowledge of the spacetime metric (technically, the stress-energy tensor can be written as the functional derivative of the matter-energy Lagrangian with respect to the metric). But, if one wants a “dynamic spacetime” in the parlance of GR, the spacetime metric must depend on the matter-energy distribution in spacetime. GR solves this dilemma by demanding the stress-energy tensor be “consistent” with the spacetime metric per Einstein’s equations16. This self-consistency hinges on divergence-free sources, which finds a mathematical counterpart in ∂∂ = 0, i.e., the boundary of a boundary principle(36). So, Einstein’s equations of GR are a mathematical articulation of the boundary of a boundary principle at the classical level, i.e., they constitute a self-consistency criterion at the classical level. In fact, our SCC will be based on the same topological maxim (∂∂ = 0) for the same reason17, as are quantum and classical electromagnetism(37). In section 5, we will show that the same structure obtains in the Maxwell action and weak field expansion of the Einstein-Hilbert action.
15 Miller showed ∂∂ = 0 applies to Regge’s discrete spacetime in Miller, W.A.: The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle. Foundations of Physics 16, 143-169 (1986).
16 Concerning the stress-energy tensor, Hamber and Williams write, “In general its covariant divergence is not zero, but consistency of the Einstein field equations demands T 0,” Hamber, H.W. and
Williams, R.: Nonlocal Effective Gravitational Field Equations and the Running of Newton’s G. arXiv: hep-th/0507017 (2005).
17 Einstein’s equations of GR are the continuous, separable, statistical approximation to the SCC of theory X.
In order to illustrate the discrete mathematical co-definition of space, time and sourcesJ, we will use graph theory a la Wise(38) and find T
1 1
, where ∂1 is a boundary operator in the spacetime chain complex of our graph satisfying ∂1∂2 = 0, has precisely the same form as the matrix operator in the discrete action for coupled harmonic oscillators. Therefore, we are led to speculate thatK T
1 1
. Defining the source vector
Jrelationally viaJ1e then gives tautologically (per the boundary of a boundary principle) both a divergence-freeJ andKvJ, where e is the vector of links and v is the vector of vertices.KvJis our SCC following from the graphical counterpart to
∂∂ = 0, i.e., ∂1∂2 = 0, and it defines what is meant by a self-consistent co-construction of space, time and divergence-free sourcesJ, thereby constrainingK and Jin Z. Thus, our SCC provides a basis for the discrete action18 in accord with Toffoli and supports our view that Eq. (3) is fundamental to Eqs. (1) & (2), rather than the converse. Conceptually, that is the basis of our discrete, graphical path integral approach to theory X. We now provide the details.
3. THE FORMALISM
3.1 The General Approach. Again, in theory X, the symmetry amplitude Z contains a discrete action constructed per a self-consistency criterion (SCC) for space, time and divergence-free sourcesJ. As introduced in section 2 and argued later in this section, we will codify the SCC usingK andJ; these elements are germane to the transition
amplitude Z in the Central Identity of Quantum Field Theory(39),
J K J J V J V K D Z 1 2 1 exp exp ) ( 2 1 exp (5).While the field is a mere integration variable used to produce Z, it must reappear at the level of classical field theory (FT). To see how the field makes it appearance in theory X, consider Eq. (5) for the simple Gaussian theory (V(φ) = 0). On a graph with N
nodes/vertices, Eq. (5) is
18 This replaces the use of classical fields to motivate the construct of QFT, as is the case in Lagrangian QFT. Wallace, D.: In defence of naiveté: The conceptual status of Lagrangian quantum field theory. Synthese 151, 33-80 (2006).
dQ dQ Q K Q J Q Z N 2 1 exp ... ... 1 (6) with a solution of
J K J K Z N 1 2 / 1 2 1 exp ) det( 2 (7). It is easiest to work in the eigenbasis of K and (as will argue later) we restrict the pathintegral to the row space ofK, this gives
1 1 2 1 1 ~ ~ ~ 2 1 exp ~ ... ~ ... N j j j j j N Q a J Q Q d Q d Z (8) where Q~jare the coordinates associated with the eigenbasis ofK and Q~Nis associated with eigenvalue zero, aj is the eigenvalue of K corresponding to Q~j, and J~jare the components of Jin the eigenbasis ofK. The solution of Eq. (8) is
1 1 2 2 / 1 1 1 1 2 ~ exp 2 N j j j N j j N a J a Z (9).On our view, the experiment is described topologically at the fundamental (graphical) level byK andJ. Again, per Eq. (9), there is no field Q~appearing in Z at this level, i.e.,
Q~is only an integration variable. Q~makes its first appearance as something more than an integration variable when we produce probabilities from Z. That is, since we are working with a Euclidean path integral, Z is a partition function and the probability of
measuringQ~k Q~o is found by computing the fraction of Z which contains Q~oat the kth graphical element(40). We have
k k o k k o k o k o k a J Q J a Q a Z Q Q Z Q Q 2 ~ ~ ~ ~ 2 1 exp 2 ~ ~ ~ ~ 2 2 (10)as the part of theory X approximated in the continuum by QFT. The most probable value of Q~oat the kth graphical element is then given by
k o k k k o k k o o k a a Q J J Q J a Q Q Q 0 ~ ~ 2 ~ ~ ~ ~ 2 1 0 ~ ~ 2 2 (11).That is, KQo Jis the part of theory X approximated in the continuum by classical FT. We note, of course, thatKQo J is in accord with acquiring classical FT from QFT via the stationary phase method (41). [The sign of the second derivative evaluated at
k k o a J Q ~ ~
goes as –ak, so this extremum is a relative maximum for positive ak (all those for K in section 4 are positive, for example).] In summary:
1. Z is a partition function for an experiment described topologically (graphically) by K J
2 1
(Figure 1).
2. Theory X gives us the probability,
Z Q Q Z Q Q k o o k ~ ~ ~ ~ , for a particular
outcome (geometric relationship under investigation) in that experiment (Figure 4).
3. KQo Jgives us the most probable values of the experimental outcomes (Figure 3), i.e., the average geometry relationally constituting the experimental equipment as it relates to the experimental procedure.
4.
Z Q Q Z Q Q k o o k ~ ~ ~ ~ andKQo J are the parts of theory X approximated in the continuum by QFT and classical FT, respectively.
3.2 The Two-Source Symmetry Amplitude/Partition Function. In order to motivate our general method, we will first consider a simple graph with six vertices, seven links and two plaquettes for our (1+1)-dimensional spacetime model (Figure 5). Our goal with this
simple model is to seek relevant structure that might be used to infer an SCC. We begin by constructing the boundary operators over our graph.
The boundary of p1 is e4 + e5 – e2 – e1, which also provides an orientation. The boundary of e1 is v2 – v1, which likewise provides an orientation. Using these conventions for the orientations of links and plaquettes we have the following boundary operator for C2 C1, i.e., space of plaquettes mapped to space of links in the spacetime chain complex: 1 0 1 0 0 1 0 1 1 0 1 1 0 1 2 (12)
Notice the first column is simply the links for the boundary of p1 and the second column is simply the links for the boundary of p2. We have the following boundary operator for C1 C0, i.e., space of links mapped to space of vertices in the spacetime chain complex:
1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 (13)
which completes the spacetime chain complex,C01 C12 C2
. Notice the
columns are simply the vertices for the boundaries of the edges. These boundary operators satisfy ∂1∂2 = 0 as required by the boundary of a boundary principle.
The potential for coupled oscillators can be written 2 1 12 2 2 2 1 , 2 1 2 1 2 1 2 1 ) , (q q k q q kq kq k q q V b a ab a b
(14) where k11 = k22 = k (positive) and k12 = k21 (negative) per the classical analogue(Figure 6) with k = k1 + k3 = k2 + k3 and k12 = –k3 to recover the form in Eq. (14). The Lagrangian is then 2 1 12 2 2 2 1 2 2 2 1 2 1 2 1 2 1 2 1 q q k kq kq q m q m L (15) so our NRQM symmetry amplitude is
Dq t Tdt mq mq V q q J q J q Z 0 2 2 1 1 2 1 2 2 2 1 ( , ) 2 1 2 1 exp ) ( (16)after Wick rotation. This gives
t k t m t m t k t m t k t m t m t k t m t k t m t k t k t k t m t m t k t m t k t m t m t k t m t k t m K 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 12 12 12 12 12 12 (17)
on our graph. Thus, we borrow (loosely) from Wise(42) and suggest K T 1 1 since 2 1 0 1 0 0 1 3 1 0 1 0 0 1 2 0 0 1 1 0 0 2 1 0 0 1 0 1 3 1 0 0 1 0 1 2 1 1 T (18)
produces precisely the same form as Eq. (17) and quantum theory is known to be “rooted in this harmonic paradigm(43).” [In fact, these matrices will continue to have the same form as one increases the number of vertices in Figure 5.] Now we construct a suitable candidate forJ, relate it toK and infer our SCC.
Recall that J has a component associated with each node so here it has components, Jn, n = 1, 2, …, 6; Jn for n = 1, 2, 3 represents one source and Jn for n = 4, 5, 6 represents the second source. We proposeJ1e, where ei are the links of our graph, since
7 6 6 5 2 5 4 7 3 3 2 1 4 1 7 6 5 4 3 2 1 1 1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 e e e e e e e e e e e e e e e e e e e e e e (19)
automatically makesJ divergence-free, i.e.,
0i i
J ,and relationally defined, e.g., vertex 1 is the origin of both links 1 and 4, and the first entry of 1e is –e1 – e4
(negative/positive means the link starts/ends at that vertex). Since Jn are associated with the vertices to represent sources, J1e is a graphical representation of “relata from relations.” [Note: 1e, which we denote v*and associate with v, is not equal to v
proper19.]
With these definitions of K and J we have, ipso facto, Kv J as the basis of our SCC since * 2 3 2 2 3 2 2 1 0 1 0 0 1 3 1 0 1 0 0 1 2 0 0 1 1 0 0 2 1 0 0 1 0 1 3 1 0 0 1 0 1 2 1 7 6 6 5 2 5 4 7 3 3 2 1 4 1 6 5 3 6 5 4 2 5 4 1 6 3 2 5 3 2 1 4 2 1 6 5 4 3 2 1 1 1 e v e e e e e e e e e e e e e e v v v v v v v v v v v v v v v v v v v v v v v v v v v T (20)
where we have used e1 = v2 – v1 (etc.) to obtain the last column. You can see that the boundary of a boundary principle holds by the definition of “boundary” and from the fact that the links are directed and connect one vertex to another, i.e., they do not start or end
19 Thus, we have characterized nodes (relata of links) in terms of the links (fundamental relations) in a non-tautological fashion as alluded to in section 1.
“off the graph.” Likewise, this fact and our definition of Jimply
0i i
J , which is our graphical equivalent of a divergence-free, relationally defined source (every link leaving one vertex goes into another vertex). Thus, the SCC Kv J and divergence-free sources
0
ii
J obtain tautologically via the boundary of a boundary principle. The SCC also guarantees that Jresides in the row space of K so, as will be shown, we can avoid having to “throw away infinities” associated with gauge groups of infinite volume as in Fadeev-Popov gauge fixing. Since K has at least one eigenvector with zero eigenvalue which is responsible for gauge invariance, the self-consistent co-construction of space, time and divergence-free sources entails gauge invariance.
Moving now to N dimensions, the Wick rotated version of Eq. (3) is Eq. (6)
dQ dQ Q K Q J Q Z N 2 1 exp ... ... 1and the solution is Eq. (7)
J K J K Z N 1 2 / 1 2 1 exp ) det( 2 Using J1e and K T 1 1 (α, β є reals) with the SCC givesKv J
, so that J K v 1
. However, K1 does not exist becauseK has a null space, therefore the row space ofK is an (N-1)-dimensional subspace of the N-dimensional vector space20. The eigenvector with eigenvalue of zero, i.e., normal to this hyperplane, is [1,1,1,…,1]T, which follows from the SCC as shown supra. Since Jresides in the row space ofK and, on our view, Z does not reflect a “sum over all paths in configuration space” but is a functional ofK and Jwhich produces a partition function for the various K J
2 1
associated with different 4D experimental configurations, we restrict the path integral of Eq. (6) to the row space ofK, i.e., Eq. (8)
20 This assumes the number of degenerate eigenvalues always equals the dimensionality of the subspace spanned by their eigenvectors, which we will see is true for K in this example.
1 1 2 1 1 ~ ~ ~ 2 1 exp ~ ... ~ ... N j j j j j N Q a J Q Q d Q d Zwhere Q~jare the coordinates associated with the eigenbasis ofK and Q~Nis associated with eigenvalue zero, aj is the eigenvalue of K corresponding to Q~j, and J~jare the components of Jin the eigenbasis ofK. Thus, our gauge independent approach revises Eq. (7) to give Eq. (9)
1 1 2 2 / 1 1 1 1 2 ~ exp 2 N j j j N j j N a J a Z Since J is defined via links we have characterized the symmetry amplitude in terms of relations and the non-zero eigenvalues ofK, which is also relational in nature.
Caveat: we chose K T 1 1
because it reproduced the action for coupled harmonic oscillators and therein V is quadratic in q. However, keep in mind that q is not
the spatial location x of a particle in the potential V as we explained above is standard in conventional NRQM, but q is the fieldvalueat a point in space per our interpretation of the path integral formalism. Thus, one must distinguish between V in the propagator of QFT’s free (Gaussian) theory and V in NRQM. We will use the free-particle propagator of QFT, which employs quadratic V per coupled harmonic oscillators, to model the twin-slit experiment since therein the NRQM ‘particle’ is free and V in the Schrödinger equation is zero. Further, at our proposed fundamental level, it is K J
2 1
that provides the basic 4D ontological depiction of the experiment and Q is merely part of the
mathematical machinery used to provide a partition function Z for K J
2 1
.
Returning to Eq. (9), we find in general that half the eigenvectors ofK are of the form x x
and half are of the form
x x
is the eigenvalue for x x
, λ + 1 is the eigenvalue for
x x , and 1 2 ,..., 0 , 2 cos 2 3 j N N j j
. The k components of xfor a given λj are
2 ,..., 1 , 1 2 cos 2 N k N k j N xjk for j > 0 and 2 ,..., 1 , 1 0 N k N x k for j = 0 (j = 0 eigenvalues of K are 0 and 2). As you can see, there are no degeneracies within the x x subspace or the x x subspace. Therefore, the only degeneracies occur between subspaces, so we know all degenerate eigenvalues are associated with unique
eigenvectors, as alluded to in a previous footnote.
We have N nodes and (3N/2 – 2) links. Define the temporal (vertical) links ei in terms of vertices vi in the following fashion:
i i i v v e 1 i = 1 to N/2 – 1 and i N i N i N v v e 2 1 2 1 2 i = 1 to N/2 – 1. Define the spatial (horizontal) links via:
i i N i N v v e 2 2 i = 1 to N/2. This gives 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 1 1 1 2 ,... 2 1 2 ,... 2 N N N i N i N i N N N N N N i N i i N e e N i e e e e e e e N i e e e e e J (21).
We then need to find the projection of Jon each of the orthonormal eigenvectors of K
that have non-zero eigenvalues. Call each projection J~i i J , where i is the ith orthonormal eigenvector. Let ai (i = 1, N-1) be the non-zero eigenvalues of K associated with the eigenvectors i , (i = 1, N-1), respectively. To complete the two-source
symmetry amplitude we need to compute the exponent
1 1 2 2 ~ N i i i a J (22)where ħ is viewed as a fundamental scaling factor with the dimensions of action. We find
Φ = (ΦS + ΦT + ΦST)/(2ħβ), where 2 2 1 2 2 2
N k N k S e N (23) involves only spatial links2 1 2 1 1 2 1 2 1 2 2 sin 2
N j N k N k k T N jk e e N (24)involves only temporal links and
2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 1 cos 2 sin sin sin 2 1 4
N j N k k N N k k k N ST N j k e N jk e e N j N j N (25)involves a mix of spatial and temporal links. Eq. (23) comes from the eigenvalue 2 associated with x x
eigenvalues associated with x x
. Eq. (24) comes from the eigenvalues associated with
x x
having omitted zero, which exists for all N under consideration.
3.3 Theory X. To summarize theory X mathematically:
J v K K J 2 1 Z
Z Q Q Z Q Q k o o k ~ ~ ~ ~ KQo J.In words, the self-consistency criterion KvJgives the actional K J
2 1
serving as a graphical model of the relations under experimental investigation, and the actional gives the partition function Z for the graph (topological level). The partition function Z gives the probability of a particular experimental outcome
Z Q Q Z Q Q k o o k ~ ~ ~ ~ , i.e., a
specific geometric relationship under investigation that comprises, in part, the
experimental arrangement (Figure 4), and the most probable of all the possible outcomes renders an average relational description (geometric level) of the experimental
arrangement KQo J, in so far as it concerns the experiment (Figure 3). Keep in mind there are relations responsible for the experimental equipment presumably not under investigation, e.g., those relations between various pieces of equipment and Mars, the experimentalist, the wall, etc. The art of good experimental procedure is to isolate the relevant results and “screen off” the irrelevant ones.
3.4 The Twin-Slit Experiment. The simple twin-slit experiment is used for a preliminary study of our two-source amplitude since our analysis reproduces the interference pattern without the use of mediating entities, such as waves or particles. We point out again that conventional NRQM uses the free-particle propagator for this case while our two-source amplitude is obtained via the discrete, free (Gaussian) theory fundamental to QFT – those are the two formalisms we relate here in order to gain insight into both. We begin with what we already know of this idealized situation per NRQM, then we make inferences concerning our graph structure via the analytic continuation of Eqs. (23) – (25).
For a free particle of mass m we have from NRQM(44) exp 2 2 exp 2 exp 2 exp 2 exp 2 2 2 2 2 imv t ipvt iv t t t imx t imx it m A (26)
where vφ is the phase velocity and equal to half the particle velocity(45) and
ψ(x,0) = Aδ(x = 0). [The conventional NRQM path integral produces a propagator and Eq. (26) is obtained from it by connecting a point Source to a point at the detector, each of these points is understood to be half of our source vectorJ, thus our use of the two-source symmetry amplitude.] Using Eq. (26), the twin-slit interference pattern is given by
2 cos 2 2 2 exp 2 exp 1 2 2 2 1 2 2 1 t t v t iv t iv (27)and therefore maxima occur at angles where
t t n v 1 2 nєintegers (28). For photons(46)
